method and calculator for modeling non-equilibrium spin polarized charge transport in nano-structures

ABSTRACT

A method and calculator for obtaining spin polarized quantum transport in 3-dimensional atom-scale spintronic (spin electronics) devices under finite bias voltage, based on implementing Density Function Theory (DFT) in combination with the Keldysh non-equilibrium Greens function (NEGF) formalism to calculate spin polarized quantum transport in 3-dimensional nanostructures under finite bias and external voltage.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority on U.S. provisional application No.60/885,056, filed on Jan. 16, 2007. All documents above are hereinincorporated by reference.

FIELD OF THE INVENTION

The present invention relates to atomistic quantum transport. Morespecifically, the present invention is concerned with a method and acalculator for calculating spin polarized quantum transport in3-dimensional nanoelectronic devices under finite bias voltage. Alldocuments below are herein incorporated by reference.

BACKGROUND OF THE INVENTION

Spin-based electronics is an emerging disruptive technology withenormous potential. A promising application is the field of computermemories, where Magnetoresistive Random Access Memories (MRAM) couldreplace the present memory technologies.

The rapid progress in the miniaturization of semiconductor electronicdevices leads to a degree of integration where quantum effects will takeprecedence over classical electronic effects. Electron spin being onesuch quantum effect, Spintronics—short for spin-basedelectronics—involves devices where both the charge and spin of anelectron are used to carry and manipulate information. Applications forspintronics-based devices include Giant Magnetoresistance (GMR),Magnetic Sensors, Magnetoresistive Random Access Memories (MRAM), SpinTransistors and Quantum Computing for example.

GMR represents a quantum technology that was successfullycommercialized, with applications in the data storage industry,principally for hard disk drives. Today almost all hard disk drives arebased on this technology.

MRAM has the potential to revolutionize and take over the memory market.MRAM technology is based on the use of magnetic moments, rather than onelectric charge, to determine the on-off state of the memory bit cell.MRAM combines the best attributes of three major existing memorytechnologies: density of DRAM (dynamic RAM), the speed of SRAM (staticRAM) and the nonvolatility of flash memory.

Magnetic sensors have the potential to be directly fabricated on siliconsubstrate, leading to highly efficient systems where sensing and logicfunctions are closely integrated. In spin-based quantum computers thathave been conceptually envisioned, the fundamental unit to representinformation is quaternary (rather than binary), which opens the door tonew computing techniques especially well adapted to deal with non-linearproblems and suitable for massively parallel computing applications.Magnetic sensors known as Anisotropic MagnetoResistance (AMR) sensorscan be presently manufactured at a reasonable cost, providing arrays ofsensors built-in on a silicon substrate. These find applications inmagnetic biosensing, non-destructive testing, position sensing, documentvalidation and magnetic imaging. Due to a number of technicalshortcomings caused by the AMR effect exploited in these sensors,magnetic tunnel junction (MJT) sensors may progressively replace AMRsensors. Since MRAMs are MJT devices, the development of MJT sensorswill be closely tied to the advances made for MRAMs.

As for spin-based transistors and quantum computing, these technologiesare still at their infancy, and further research and development areneeded before commercial applications become available. Spin-basedtransistors offer an alternative for smaller energy consumption andfaster performance.

It is expected that GMR will continue to be a driving market force forspintronics until MRAM takes off and becomes widely commerciallyavailable. In the short term, the hard-disk market is expected tocontinue to be a primary driver for technological advancement, pulled bya constant need to increase the data storage capacity that leads todevices with increased surface density. With already some initialsuccess for applications in the space and military domains, MRAM is justentering the commercial market. Prototypes of MRAM-based products arepresently available and it is expected that commercially viable productswill be soon available. Continuous research and development activitiesin the fields of magnetic device elements, magnetic materials andprocess integration for magnetic components will allow the technology tofurther develop and to eventually become widely commercially available.In the long term, MRAM may become a memory for many applications, sinceit is at the same time nonvolatile, durable, fast and dense, and thuseliminates the need to combine memories of different types. It has thepotential to be used for a wide range of applications such as mobilephones, digital cameras, game consoles, personal digitalassistants/organizers, personal computers, automobiles, etc.

The broad adoption of spintronics depends on a number of restrainingfactors that still constitute considerable challenges, at least in theshort term. One first such factor relates to the production ofsemiconductor devices with new materials. Indeed, spintronics do not usethe same materials used in conventional semiconductors. Sincespintronics devices use magnetism, metallic substrates are usuallyneeded. Materials that find applications in spintronics include nickel,iron, cobalt and their alloys and other compounds. Their use inconjunction with traditional semiconductor materials results inchallenges in terms of compatibility, integration, etching, patterning,and production. A second challenging factor is the reliability ofspintronics device, which is not demonstrated yet. A third restrainingfactor might be the fact that MRAMs requires high current for the magnetto switch between states, which causes problem for low-power operation.

Methods that can accurately model performance of modern hard-drives andmagnetic memory (MRAM) and aid in the design process and evaluation offuture technology have thus become essential. As the devices continue todownsize into the nanoscale limit, currently available modeling methodsare no longer adequate or even valid, and new methods are needed tocontinue to support research and development.

The tunnel magnetoresistance (TMR) effect in systems with spin-polarizedtransport is emerging as a basic physical principle for near-futureinformation storage technology, magnetic sensor, magnetic random accessmemory (MRAM), magnetic programmable logic, as well as spintronics (seeW. J. Gallagher et al. IBM Res. & Dev., Vol 50. 5-23A (2006)).

TMR devices typically comprise an insulating (I) tunnel barrierseparating two ferromagnetic (FM) metal layers in the form of FM-I-FM,as shown in FIG. 1 for example (W. Butler, Nature Materials, 3, 845(2004)). Crucial to the development of TMR technology is the materialand device design automation. At present, since the insulating layer (I)is only a few atomic layers thick, even a few impurities and/or defectsmay have substantial quantitative influence to device operation.Furthermore, since TMR is a quantum mechanical effect, a proper designautomation method should be based on atomistic quantum mechanical firstprinciples. Such a method should be able to predict how spin-polarizedcharge transport is related to atomic structure, and how TMR depends onexternal control parameters such as voltage and temperature.

TMR effect will be briefly described in relation to FIG. 1. It is foundexperimentally that when the magnetic moments of two ferromagneticlayers 1 and 2 are antiparallel (AP), the tunnel resistance R_(AP)across the tunnel barrier is large, i.e. the tunneling current I_(AP) issmall; whereas, when the moments are parallel (P), the tunnel resistanceR_(P) is small, i.e. the tunneling current I_(P) is large. The switchingbetween large and small resistances is therefore achieved by therelative orientation of the magnetic moments in the two FM layers.

For MRAM application, R_(AP) and R_(P) are used as “1” and “0” in anon-volatile memory device. In information storage or magnetic sensorapplication, an external magnetic field (from the magnetic bits in amedia, for example) flips one of the moments, leading to a change oftunneling current, which is then detected.

An important device parameter is the magneto-resistance ratio, definedby R_(AP) and R_(P) as: R=(R_(AP)−R_(P))/R_(P)=(I_(P)−I_(AP))/I_(AP):the greater R, the more sensitive the device becomes, which isdesirable. While magnetic tunnel junctions have been studied for manyyears, it was only very recently that large values of R have beenachieved at room temperature, leading to commercial applications of TMR.

FIG. 2 illustrates enhancements of TMR ratio at room temperature, due tothe discovery of the MgO tunnel barrier (M. Coey, Nature Materials, 4, 9(2005)). In addition to oxide barriers, there has been other research ondevices where the barrier is a single molecular layer or even a singlemolecule. These molecular TMR devices include carbon nanotubes (K.Tsukagoshi, B. W. Alphenaar and H. Ago, Nature, 401, 572 (1999).),organic semiconductors (Z. H. Xiong, Di Wu, Z. Valu Vardeny and JingShi, Nature, 427, 821 (2004)), and organic molecules (J. R. Petta, S. K.Slater and D. C. Ralph, Phys. Rev. Lett. 93, 136601 (2004)), sandwichedbetween Ni, Co and Fe materials. The TMR ratio in these moleculardevices is typically around 10-20% at present, pointing to substantialroom for further improvements.

The rapid progress in TMR technology, as exemplified in FIG. 2, is inpart due to better material preparation and control. It is also directlyrelated to an increased understanding of the basic physics of spinpolarized quantum transport. Indeed, the theoretical prediction (W. H.Butler et al. Phys. Rev. B 63, 054416 (2001)) of coherent tunnelingcoupled to symmetry of electron wave function in Fe—MgO interfacecontributed to the experimental discovery of the large TMR.

In order to rapidly develop TMR technology to large-scalecommercialization, an atomistic modeling method of quantum transport isneeded for assisting experimental work. This is especially true due tothe fact that there still exist many experimental facts that have notbeen understood. For instance, while theoretical prediction (W. H.Butler et al., Phys. Rev. B 63, 054416 (2001)) of TMR ratio forFe—MgO—Fe device can be greater than several thousands, experimentalresults (as exemplified in FIG. 2) are still considerably lower. Thereason may be related to the quality of the atomic arrangement at theFe/MgO interface, the existence or absence of oxygen vacancies and/orother defects. More importantly, much experimental data (S. S. P. Parkinet al. Nature Materials, 3, 862 (2004); S. Yuasa et al., NatureMaterials, 3, 869 (2004)) show that the TMR ratio is monotonicallydecreased by applied bias voltage and reduces to zero when the voltageis about 0.5-1.0 volt. Published theoretical work to date predicted asubstantial increase of TMR by bias for systems with asymmetric atomicstructure (C. Zhang et al., Phys. Rev. B 69, 134406 (2004)). It is clearthat all the modeling methods presented in literature, while useful foracademic research, are not capable of making quantitative predictions ofspin-polarized quantum transport for practical TMR devices at theatomistic level under nonequilibrium conditions.

Theoretical modeling of spin-polarized transport including atomic andmaterial properties is a very difficult problem.

In DFT, as described in the art, the Hamiltonian operator Ĥ of a systemis determined as a functional of a local electron charge density ρ(r),i.e. Ĥ=Ĥ[ρ(r)].

In a transport problem, the system has open boundaries connecting toelectrodes and operates under external bias and gate potentials, whichdrive the device to non-equilibrium: in other words, theenvironmental-group of the system comprises one or more electrodes andpossibly metallic gates and substrates where the device is embedded, andthe device-group is the electronic device scattering region, whichcomprises at least one atom. The charge density ρ(r) is thus to bedetermined under such conditions. Obtaining Ĥ and ρ(r) is aself-consistent process, wherein Ĥ is obtained from ρ(r), and then,using Ĥ, ρ(r) is evaluated, in an iterative process until Ĥ converges.The nonequilibrium device conditions may be accounted for by using theKeldysh non-equilibrium Green's function (NEGF) for example, toconstruct ρ(r) from Ĥ.

Indeed, as known in the art, the NEGF-DFT formalism is able to calculatecharge density ρ(r) for open quantum device systems under a bias voltageentirely self-consistently without depending on using phenomenologicalparameter. Since the charge density ρ is constructed from NEGF, thenon-equilibrium nature of the device can be handled properly. Atoms inthe device scattering region and in the electrodes are treated at equalfooting, therefore allowing a realistic electrodes and contactsmodeling. NEGF treats the discrete and the continuum parts of theelectron spectra at equal footing, so that all electronic states areincluded properly into the calculation of the device Hamiltonian H.

It is to be noted that NEGF-DFT has already been applied to devices withsizes and complexities no other self-consistent atomistic formalism ofthe art could handle. However, it still cannot be used for spinpolarized charge transport in 3-dimensional magnetic devices.

The existing atomistic modeling methods for TMR devices can be roughlycategorized into three classes:

-   -   (i) Non-self-consistent tight binding methods can account for        some qualitative features of transport, but they depend on        semi-empirical parameters thereby having limited predictability        (H. Mehrez et. al. Phys. Rev. Lett. 84, 2682 (2000); S.        Krompiewski, R. Gutierrez, and G. Cuniberti, Phys. Rev. B 69,        155423 (2004); E. G. Emberly and G. Kirczenow, Chem. Phys. 281,        311 (2002).).    -   (ii) Layer-KKR method with DFT is a self-consistent method for        electronic structure (W. H. Butler et al., Phys. Rev. B 63,        054416 (2001); C. Zhang et al., Phys. Rev. B 69, 134406        (2004); J. M. MacLaren et. al., Phys. Rev. B 59, 5470 (1999)).        After DFT is numerically converged, transport properties are        computed by applying atomic sphere approximation (ASA). ASA is,        in fact, an “art” and the results often depend how it is applied        (For example, it has been shown that calculated tunneling        conductance of Fe—GaAs interface sensitively depends on how ASA        is applied, making quantitative predictions very difficult to        make. K. Xia, private communication (2005)). This method has        difficulty when there is an external bias voltage, for instance        it predicted an increase of TMR ratio rather than the        experimentally observed decrease as the bias is increased (C.        Zhang et al., Phys. Rev. B 69, 134406 (2004)). Finally, the        method requires very large computational time; and it is unclear        if a bias and gate voltage can be modeled. This method is        limited for application in systems consisting of layers of        materials.    -   (iii) LMTO method has also been successfully applied to        investigate TMR systems at zero bias voltage (K. M. Schep, et        al., Phys. Rev. B 56, 10805 (1997); K. Xia, et al. Phys. Rev. B        63, 064407 (2001)). It is also based on ASA, and has difficulty        in applying to non-space filling situations such as molecular        TMR structures. In addition, at present no LMTO methods for TMR        device simulation has and can include a bias voltage in the        required self-consistent manner. Present implementations of this        method can only be applied to bulk materials or layered bulk        materials without external bias or gate voltage.

The above existing methods prove to be limited in their applicationdomain. In particular, they have only been applied to bulk systems atzero external potential.

Therefore, there is a need for a state-of-the-art atomistic quantumtransport modeling method and associated method that is capable offilling the gap of TMR modeling for industrial applications.

The present description refers to a number of documents, the content ofwhich is herein incorporated by reference in their entirety.

SUMMARY OF THE INVENTION

More specifically, in accordance with the present invention, there isprovided a method for calculating spin polarized quantum transport in a3-dimensional nanoelectronic device under non-equilibrium conditions atfinite bias voltage, comprising the steps of: a) self-consistentlyobtaining the Hamiltonian H of the device using DFT within the standardlocal spin density approximation for open device structures; b)constructing a non-equilibrium density matrix of the device usingKeldysh non equilibrium Green's functions in spin space; and c)calculating spin-dependent transmission coefficients from the Green'sfunctions, wherein step b) comprises a transverse momentum sampling ofthe Brillouin zone for contribution of each transverse Bloch state indirections perpendicular to the current flow for converging the densitymatrix

There is further provided a method for self-consistently computingnon-equilibrium spin resolved density matrix for three-dimensional (3D)magnetic systems consisting of a central scattering region comprising anon-magnetic material, a left and a right magnetic electrodes serving asdevice leads which connect the device to the outside world, comprisingself-consistently calculating the Hamiltonian of each region within DFTin the local spin density approximation and generalized gradientapproximation under a potential including an applied bias potentialV_(b) that drives the current flow; whereby the Hamiltoniancorresponding to the left and right ferromagnetic electrodes arecalculated as isolated bulk material and the k_(∥)-dependent retardedself-energies of each lead are determined; and the Hamiltoniancorresponding to the central region is calculated self-consistentlyusing the non-equilibrium electron density matrix, calculated over a 2D(in the x-y direction) Brillouin zone for contributions of eachtransverse Bloch state; and spin currents are calculated by integratingthe contributions from each transverse Bloch state.

There is further provided a method for calculating spin polarizedquantum transport in a 3-dimensional nanoelectronic device undernon-equilibrium conditions at finite bias voltage, comprising the stepsof: self-consistently obtaining the Hamiltonian H for an open deviceusing DFT; obtaining the electronic structure of the device by usingNEGF to handle the nonequilibrium statistics, and the device transportboundary conditions using a real space numerical technique; andimplementing local spin density functional and generalized gradientapproximation) for nonequilibrium NEGF-DFT calculation.

There is further provided a computer system for calculation ofspin-polarized charge current and spin current transport in magneticdevices; the system comprising a calculator; and a database; wherein thecalculator computes the electronic structure of each 3D magneticelectrode of the device using DFT at equilibrium; and stores them in thedatabase; then, the calculator shifts the electrode potential by anamount determined by an external bias voltage; for a first magneticconfiguration of the electrodes: i) the calculator constructs a firstdevice Hamiltonian H from a first spin-resolved nonequilibrium densitymatrix and, using the first H, the calculator computes the NEGF of thescattering region of the device for the first magnetic configuration ofthe electrodes; ii) from the obtained NEGF, the calculator calculates asecond spin-resolved nonequilibrium density matrix; and using the secondspin-resolved nonequilibrium density matrix, the calculator constructs asecond device Hamiltonian; the calculator repeating steps i) and ii)until the second device Hamiltonian H and the first device Hamiltoniandefer by only a pre-specified amount; then the calculator computestransport properties for the device in the first magnetic configurationof the electrodes; and then changes the first magnetic configuration ofthe electrodes to a second magnetic configuration of the electrodes andstarts again.

There is further provided a computer system for calculating spinpolarized quantum transport in a 3-dimensional nanoelectronic deviceunder non-equilibrium, comprising: a calculator; a controller; and astorage unit; wherein the calculator, from a periodic atomic structurein 3-dimensional space of each electrode of the device, computes theelectronic structure thereof; the storage unit storing the electronicstructure; and the controller then shifting the electrode potential byan amount determined by an external bias voltage; for a first magneticconfiguration of the electrodes, the calculator constructs a firstdevice Hamiltonian H from a first spin-resolved nonequilibrium densitymatrix; using the first device Hamiltonian H, the calculator computesthe NEGF of a scattering region of the device for the first magneticconfiguration of the electrodes; from the obtained NEGF, the calculatorobtains a second spin-resolved nonequilibrium density matrix; using thesecond spin-resolved nonequilibrium density matrix, the calculatorconstructs a second device Hamiltonian; until the second deviceHamiltonian and the first device Hamiltonian defer by a pre-specifiedamount; and then the calculator computes transport properties transportproperties for the device in the first magnetic configuration; then, thecontroller changes to a second magnetic configuration of the electrodes;and for the second magnetic configuration, the calculator starts again.

Other objects, advantages and features of the present invention willbecome more apparent upon reading of the following non-restrictivedescription of specific embodiments thereof, given by way of exampleonly with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

In the appended drawings:

FIG. 1 is a schematic plot of a FM-I-FM TMR device as known in the art;

FIG. 2 shows a large TMR ratio at room temperature of MgO based TMRdevices;

FIG. 3 is a schematic plot of the atomic structure of a Fe/MgO/Femagnetic tunnel junction (MTJ);

FIG. 4 is a flowchart of a method according to an embodiment of anaspect of the present invention;

FIG. 5 show spin-polarized transport calculations of Ni—BDT-Ni molecularTMR device (FIG. 5 a) using the present method; FIG. 5 b: TMR ratioversus bias voltage for hollow site binding—Inset: TMR for bridge sitebonding;

FIG. 5 c: I-V curves for PC set up of the leads' magnetization—Solidline: total current; dashed line: I_(↑); dotted line: I_(↓); FIG. 5 d:I-V curves for APC (antiparallel configuration) setup; Inset:spin-injection coefficient η coefficient versus bias voltage V_(b)—solidline: APC; dashed line: PC (parallel configuration); and

FIG. 6 shows calculation of spin-polarized quantum transport inFe—MgO—Fe MTJ devices using the present method; FIG. 6 a: schematicalillustration of the MTJ—Figures b) and c) I-V curves for 5-layer PC andAPC, respectively. diamonds: total current; squares: I_(↑) circles:I_(↓); insets: I-V curves for small ranges of V_(b); FIG. 6 d: TMR vs.bias V_(b) for a 5-layer device (diamond).

DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The present invention is illustrated in further details by the followingnon-limiting examples.

There is provided a method and a calculator for calculating spinpolarized quantum transport in 3-dimensional nanoelectronic devicesunder finite bias voltage. The method and calculator perform firstprinciples calculations of spin polarized quantum transport in3-dimensional atomic-scale spintronics (spin electronics) devices underfinite bias voltage (externally applied electric field).

Generally stated, the present method and calculator are based onimplementing Density Function Theory (DFT) in combination with Keldyshnon-equilibrium Green's functions (NEGF) within the Local Spin DensityApproximation (LSDA) and Generalized Gradient Approximation (GGA). Thelocal spin-density approximation (LSDA) is a generalization of thelocal-density approximation (LDA), in which the functional depends onlyon the density at the coordinate where the functional is evaluated, toinclude electron spin. Generalized gradient approximations (GGA) arestill local but also take into account the gradient of the density atthe same coordinate.

In the present method, Keldysh non-equilibrium Green's functions (NEGF)are formulated in spin space so that the non-equilibrium density matrixcan be evaluated for different spin channels. Then, in order to obtainthe correct bulk as well as surface magnetism (for magnetic materialswhich serve as the device electrodes) for a magnetic device, carefultransverse momentum sampling of the Brillouin zone in directionsperpendicular to the current flow is performed for converging thedensity matrix. Denser k sampling, typically several hundred to athousand points, and some times up to tens of thousands, is necessary toconverge the NEGF.

In the present method, the NEGF-DFT is used to account for themicroscopic details of the atomic structure and electronic states atnon-equilibrium, as follows. DFT is used to self-consistently calculatethe Hamiltonian H of the device. After the device Hamiltonian H iscalculated by DFT, the electronic structure of the system is obtained.Quantum transport being a non-equilibrium problem since a current isflowing, one has to populate the electronic structure by propernon-equilibrium statistics, which is accomplished by using NEGF. To dealwith the device transport boundary conditions and externally appliedbias voltages, a real space numerical technique is needed. The resultingDFT formalism thus allows calculating quantum transport properties ofany atomic structure from first principles without any phenomenologicalparameter. In order to analyze spin-polarized quantum transport in TMRdevice further, local spin density functional (LSDA) and GeneralizedGradient Approximation (GGA) are then incorporated. A difficulty arisesfrom the huge number of Bloch bands (Brillouin zones) in the magneticlayer, which is therefore sampled. Finally, in order to obtain accuratebulk and surface magnetism, the quality of wave function basis sets andpseudo potentials is strictly controlled.

The method and calculator of the present invention are able toself-consistently compute non-equilibrium spin resolved density matrixfor three-dimensional (3D) magnetic systems with a two—(as shown in FIG.3 for example) or multi-probe device setup, so that transport can becalculated for both parallel and anti-parallel magnetizationconfigurations of the contacts of an MNM device, where M stands formagnetic materials (e.g. Fe, Ni, Co, etc) and serves as device leadswhich connect the MNM device to the outside world; and N stands fornon-magnetic materials that can be a metal (e.g. Cu etc), molecules(e.g. carbon nanotubes, organic molecules, etc), semiconductors (e.g.GaAs etc), or insulating materials (e.g. MgO etc).

The present method will be described in relation to a Fe/MgO/Fe magnetictunnel junction (MTJ) shown in FIG. 3 as an illustration of the kind ofmagnetic devices of interests.

This magnetic tunnel junction (MTJ) may be characterized as composed ofthree parts: 1) a central scattering region which consists of MgO layersand a few layers of the Fe atoms on either side of the MgO; 2) a left Feelectrode which extends to z=−∞. In the transverse direction, namely x-ydirection, the electrode also extends to infinity so that the Fe form aninfinitely large periodic structure; and 3) a right Fe electrode with asimilar structure as that of the left electrode.

The left and right electrodes do not have to be the same material.

It is to be noted that for magnetic systems, three-dimensional (3D)leads are necessary to describe both the surface and bulk magnetism ofthe contacted electrodes realistically.

Along the transport direction (z axis), the two ferromagnetic leadsextend to reservoirs at z=±∞. The central scattering region is chosensufficiently large in the z direction such that: (i) the potentialsoutside the central region are taken as equivalent to bulk and (ii) thecentral scattering region is sufficiently large such that the matrixelements coupling the left and right leads are zero. The electrochemicalpotentials of the left and right leads, μ_(L) and μ_(R), are given bythe bulk Fermi level of the ferromagnets that can be calculated by DFTat equilibrium, and the applied external bias voltage.

Because the device is x-y periodic, the Eigenstates of the system can bewritten in terms of their transverse momentum:

Ψ^(k) ^(∥) (R _(∥) +r)=e ^(ik) ^(∥) ^(·R) ^(∥) ^(.) ×e ^(ik) ^(∥)^(·r.)Φ^(k) ^(∥) (r)

where k_(∥) is a Bloch wave vector, R_(∥)=n_(x)a+n_(y)b is a latticevector, and Φ^(k) ^(∥) is the x-y periodic Bloch function.

The Keldysh non-equilibrium Green's functions (NEGF) are used tocalculate electron density matrix at non-equilibrium and transportproperties of the system. In particular, the retarded Green's functionin k_(∥)space corresponding to the left-central-right regions of thedevice is obtained. The coupling of the left and right regions to theremaining part of the semi-infinite electrodes is fully taking into

account by the self-energies

$\sum\limits_{L}^{k_{//}}\mspace{14mu} {{and}\mspace{14mu} {\sum\limits_{R}^{k_{//}}.}}$

To analyze spin-polarized transport, these matrices are extended intospin space. Each matrix element thus becomes a two-by-two matrix, whichspecifies spin-up, spin-down and the connection between the two spinspaces. There is no restriction of spin colinearity, hence the left andright leads (and possibly any other part of the system) can havearbitrary relative magnetic orientation. For problems such as spintransfer torque, anti-ferromagnet tunnel junction and spin-orbitalinteraction, the calculation capability of noncolinear spin isimportant. Below, non-colinearity is not considered and all theformulations below are for colinear systems.

The Hamiltonian of each region is calculated self-consistently withinDFT by solving the Kohn-Sham equation:

${\left\lbrack {{- \frac{\nabla^{2}}{2}} + {\int{{r^{\prime}}\frac{\rho \left( r^{\prime} \right)}{{r - r^{\prime}}}}} + {V_{ext}(r)} + {V_{xc}(r)}} \right\rbrack {\Psi (r)}} = {ɛ_{i}{\Psi (r)}}$

where ρ(r) is the total electron density, V_(XC) is theexchange-correlation functional and V_(ext) is any potential includingthe pseudopotential that defines the atomic core and the applied biaspotential V_(b) that drives the current flow.

The spin-dependent exchange-correlation potential is treated at thelocal spin density approximation, where one distinguishes spin-up andspin down densities ρ^(↑) and ρ^(↓) and the total density is given byρ=ρ^(↑)+ρ^(↓).

When the central scattering region includes enough layers of theferromagnetic lead atoms, the electronic structure of the left and rightregions can be safely considered as that of bulk, which can becalculated with a supercell DFT analysis. In other words, the parts ofthe Hamiltonian corresponding to the left and right ferromagneticelectrodes are calculated as isolated bulk material whose electrondensity is given by the Kohn-Sham single-particle states:

${\rho (r)} = {\sum\limits_{i}{{f\left( E_{i} \right)}{\Psi_{i}}^{2}}}$

where f(E) is a Fermi-Dirac distribution.

Using the Fermi-Dirac distribution, it is assumed that the left andright ferromagnetic leads are in equilibrium contact with theircorresponding reservoirs. The k_(∥)-dependent retarded self-energies ofeach lead are determined. In constructing the self-energies andpotential matrices for each lead, the reference spin is rotated tospecify the relative magnetic orientation (for example parallel orantiparallel) of the two ferromagnetic leads.

The parts of the Hamiltonian corresponding to the central region arecalculated self-consistently using the non-equilibrium electron densitymatrix, calculated over the 2D (in the x-y direction) Brillouin zone(BZ) for contributions of each transverse Bloch state. Finally, the spincurrents (spin-polarized charge currents) are calculated by integratingthe contributions from each transverse Bloch state using the Landauerformula:

$I_{\sigma} = {\frac{2^{2}}{h}{\int_{BZ}{{k_{//}}{\int{{{ɛ\left\lbrack {{f\left( {ɛ - \mu_{L}} \right)} - {f\left( {ɛ - \mu_{R}} \right)}} \right\rbrack}} \times {T_{\sigma}\left( {ɛ,k_{//}} \right)}}}}}}$

where each quantity in the right-hand side is defined in spin space,with the spin index in the transmission coefficient and the current.

As people in the art will appreciate, the introduction of x-yperiodicity dramatically increases the number of calculation stepscompared to a 1D device. First, there are significantly more interactingunit cells in the x-y direction, even with nearest-cell interaction.Second, there is an additional k integration for the calculation of thedensity matrix and the current. For magnetic tunnel junctions, a hugenumber of k_(∥) is necessary to converge the results.

Therefore, in contrast with methods for 1D devices, where there is noBrillouin zone (BZ), the present method comprises computing thescattering of incoming states from the 3D leads, by covering the entire2 dimensional x-y Brillouin zone (BZ) through careful k-sampling.Because of BZ, the density matrix becomes functions ofk_(x)−k_(y)=k_(∥), and for each values of k_(x)−k_(y), the methodcomputes the density matrix and then adds them up, using DFTself-consistent loop. Thus, for example, to compute 100*100 k_(x)−k_(y)values, 10,000 density matrices are needed and added up to obtain thefinal density matrix for one energy value. Then a large number of energyvalues are computed. The process is repeated during the self-consistentloop in the NEGF-DFT. These is extremely time consuming, and that's whya parallel computation is done.

Because of BZ, transmission function becomes extremely complicated ink_(x)−k_(y) space. Namely, in the step 130 below, the k_(x)−k_(y)dependence is extremely complicated for spin transport and one must takegreat care to calculate this dependence accurately.

Because of BZ, all quantities become functions of k_(x)−k_(y), forexample the self-energy. Thus all of them become more difficult tocalculate for similar reasons as above.

To expand the electronic wave functions and construct the matrixelements of the retarded Green's function in k_(∥)space, an atomicorbitals basis set is used for the DFT, wherein the atomic cores aredefined by standard norm-conserving non-local pseudopotentials. Specialcare must be given to the pseudopotentials and basis sets in order toobtain an accurate description of the band structure near the Fermilevel, which is particularly important in studying spin-polarizedtransport. On the one hand, the calculation of the Green's functionrequires a reasonable sized basis set, while on the other hand a smallbasis set does not give accurate results. Therefore a reasonablecompromise should be adopted. Pseudopotentials and basis sets thataccurately reproduce the electronic structure of the electrode andbarrier material are found no to necessarily reproduce the electronicstructure of the more complicated electrode/barrier interface.Therefore, these inputs are carefully constructed to accuratelyreproduce electronic structures of the bulk materials and interfaces.

Therefore for DFT, standard norm-conserving pseudopotentials are usedand numerical orbitals are used as a basis set. Fine real spacediscretization is used. In the NGEF-DFT self-consistent calculation ofthe density matrix, a complex energy contour integration with a numberof points is necessary at zero bias, plus energy points along realenergy axis in the bias window when V_(b) is nonzero. A number of kpoints are used to sample the Brillouin zone for each energy points forconverging the density matrix. Dense k sampling is necessary to obtainaccurate results.

As generally summarized in FIG. 4, the present method comprisesself-consistently calculating the electronic Hamiltonian of the deviceby using DFT within the standard local spin density approximation (step110); constructing the non-equilibrium density matrix using NEGF so thatthe open device structure and the nonequilibrium transport conditionsare taken into account (Step 120). After the NEGF-DFT self-consistentiteration is converged, the spin-dependent transmission coefficients arecalculated from the Green's functions as known in the art (step 130):

$\begin{matrix}{{{{T_{\sigma}\left( {E,V_{b}} \right)} = {\sum\limits_{k_{x},k_{y}}{T_{\sigma}^{k_{x},k_{y}}\left( {E,V_{b}} \right)}}},{where}}{T_{T_{\sigma}}^{k_{x},k_{y}} \equiv {{Tr}\left\lbrack {{{Im}\left( \sum\limits_{L}^{r} \right)}G^{r}{{Im}\left( \sum\limits_{R}^{r} \right)}G^{a}} \right\rbrack}}} & (1)\end{matrix}$

is the transverse momentum resolved transmission coefficient.

All quantities in Equation (1) are functions of transverse momentum.Here σ=↑,↓ is the spin index; G^(r,a) are the retarded (advanced)Green's function matrices in spin or orbital space; and

$\sum\limits_{L,R}^{r}$

are the retarded self-energies due to the existence of the bulk −3D left(right) leads. T_(σ)(E, V_(b)) depends on V_(b) because the quantitieson the right-hand side of Equation (1) follow from the V_(b) andspin-dependent self-consistent Kohn-Sham potentials.

Finally, the spin current (spin-polarized charge current) is obtained by(see Landauer formula hereinabove):

${I_{\sigma}\left( V_{b} \right)} = {\frac{e}{h}{\int_{\mu_{L}}^{\mu_{R}}{{{T_{\sigma}\left( {E,V_{b}} \right)}\left\lbrack {{f_{L}\left( {E - \mu_{L}} \right)} - {f_{R}\left( {E - \mu_{R}} \right)}} \right\rbrack}{E}}}}$

where μ_(R,L) is the electrochemical potential of the left (right) leadsand μ_(L)−Ξ_(R)=eV_(b); ƒ_(L,R)(E−μ_(R,L)) are the Fermi functions

The total charge current is given by I≡I_(↑)+I_(↓).

TMR is obtained from the total currents asTMR=(I_(parrallel)−I_(antiparrallel))/I_(antiparrallel).

In step 110, for DFT, standard norm-conserving pseudopotentials are usedand numerical orbitals are used as a basis set. Fine real spacediscretization is used.

In step 120, effects of an external bias voltage V_(b) is included inthe NEGF so that the density matrix is a function of V_(b). In theNGEF-DFT self-consistent calculation of the density matrix, a complexenergy contour integration with a number of points is necessary at zerobias, plus energy points along real energy axis in the bias window whenV_(b) is nonzero. A number of k points are used to sample the Brillouinzone for each energy points for converging the density matrix. Denser ksampling is necessary to obtain accurate results.

FIG. 5 show spin-polarized transport calculations of a Ni—BDT-Nimolecular TMR device using the present method. A Ni-1,4-benzenedithiolate (BDT)-Ni device was used as an example, asschematically shown in FIG. (5 a) in which the BDT is in contact withtwo semi-infinite Ni (100) surfaces having periodic x-y extent.

For the Ni—BDT-Ni device, the TMR ratio is found to decline from about27% at small bias voltages to essentially zero at bout 0.5 V (FIG. 5 b).Such a quench of TMR ratio by bias voltage is also seen in experimentaldata although the experimental device involved more than one molecule(K. Tsukagoshi, B. W. Alphenaar and H. Ago, Nature, 401, 572 (1999); Z.H. Xiong, Di Wu, Z. Valu Vardeny and Jing Shi, Nature, 427, 821 (2004);J. R. Petta, S. K. Slater and D. C. Ralph, Phys. Rev. Lett. 93, 136601(2004).).

As shown in FIGS. 5 c and 5 d, as a function of bias, the spin currentis found to vary nonlinearly: up and down spin currents can be larger orsmaller than each other, and a change of sign of spin-injection occurs,due to transport resonances mediated by a combination of molecularstates and the Ni surface electronic structure.

Therefore, the voltage decay scale of the TMR (FIG. 5 b), the TMR ratio(FIG. 5 d), and the I-V curves (FIGS. 5 a,c) are all very similar todata measured in existing molecular spintronics devices.

These and other results obtained by the present NEGF-DFT-LSDA (GGA)method are proof that the present method has quantitatively captured theessential physics of TMR devices.

There is further provided a calculator, implementing the method of thepresent invention. In an embodiment, this calculator has beenimplemented in Matlab™, with time critical portions of the code writtenin Java™, under the appellation MATDCAL (Matlab™-based DeviceCalculator). In order to facilitate parallel computing within Matlab™, ahigh-performance parallel computing toolbox for Matlab™ was developed bywrapping a third-party implementation of MPI2.0 libraries.

MATDCAL has a large set of features, including the following: 1)electronic and magnetic structure calculations of bulk systems andXY-periodic two-probe devices; 2) band structure calculations of bulksystems; 3) equilibrium and non-equilibrium spin-dependent densitymatrix and density of states calculations; 4) spin-dependenttransmission, spin-current, and IV-curve calculations of XY-periodictwo-probe devices under finite bias voltage; 5) spin-dependentscattering states of two-probe devices under finite bias voltage; and 6)a comprehensive analysis library for extracting spin polarized quantumtransport features from the NEGF-DFT/LSDA/GGA calculations, as well asmany analysis tools for calculating other quantum transport features ofnanoelectronic devices such as capacitances, time dependent transients,nonlinear conductance coefficients, admittance, impedance, etc.

In addition to calculating spin-dependent characteristics, MATDCAL alsoimplements the regular DFT-NEGF formalism, and is thus capable ofnon-spin electron quantum transport in nanostructures. MATDCAL has beenextensively tested and can now be considered a mature research code. Ithas already been used to perform a state-of-the-art calculation of theTunneling Magnetoresistance Ratio (TMR) and spin currents in a MagneticTunnel Junction (MTJ) under finite bias voltage. (“Nonlinearspin-current and magnetoresistance of molecular tunnel junctions”, DerekWaldron, Paul Haney, Brian Larade, Allan MacDonald and Hong Guo, Phys.Rev. Lett. 96, 166804 (2006). “First principles modeling of tunnelmagnetoresistance of Fe/MgO/Fe trilayers”, Derek Waldron, VladimirTimoshevskii, Yibin Hu, Ke Xia and Hong Guo, Phys. Rev. Lett. 97, 226802(2006); “Ab initio simulation of magnetic tunnel junctions”, DerekWaldron, Lei Liu and Hong Guo, Nanotechnology, 18, 424026 (2007)).

The present calculator quantitatively and accurately modelsspin-polarized charge current and spin current transport in magneticdevices from atomic microscopic point of view. The magnetic devices arein the form of MNM, as described hereinabove.

So far only the present calculator has been able to accomplish this 3Dsimulation of such magnetic devices.

FIG. 6 shows calculation of spin-polarized quantum transport in aFe—MgO—Fe MTJ device (Figure a) using the present calculator. FIGS. 6(b) and (c) plot the calculated current-voltage (I-V) characteristics(solid line) for a five-layer MgO device in PC and APC, respectively.FIG. 6 d shows a decrease of TMR ratio with a voltage scale of about0.5-1.0 V, very similar to measured data (S. S. P. Parkin et al. NatureMaterials, 3, 862 (2004); S. Yuasa et al., Nature Materials, 3, 869(2004)). In addition, the gap in I-V curve is about one volt, also inagreement with the measured results (W. Wulfhekel, et al., Appl. Phys.Lett. 78, 509 (2001)). All the obtained voltage scales for transportfeatures are consistent with experimental data.

For the 3D left magnetic electrode, which is a periodic atomic structurein 3-dimensional space, the calculator computes its electronic structureusing DFT at equilibrium. The electronic structure (potential, overlapmatrix, Hamiltonian matrix, etc) is stored into a left-lead database.The same is done for the 3D right magnetic electrode and stored in aright-lead database.

Then the calculator shifts the electrode potential by an amountdetermined by the external bias voltage. The electrostatic potential ofthe scattering region is determined by the NEGF-DFT self-consistentprocedure.

For a given magnetic configuration of the two electrodes (say they areparallel), the calculator starts from any initial spin-resolvednonequilibrium density matrix and construct an initial deviceHamiltonian H.

Using the H, the calculator computes the NEGF of the scattering regionof the device for the given magnetic configuration of the twoelectrodes. From the obtained NEGF, the calculator calculates a newspin-resolved nonequilibrium density matrix.

Using this spin-resolved nonequilibrium density matrix, the calculatorconstructs a new H. The calculator repeats this process until“self-consistency”. Here, self-consistency means the new H and the old Hdefer by only a tiny amount that is pre-specified.

When self-consistency is obtained, the calculator proceeds to computetransport properties for the device in the specified magneticconfiguration of the two electrodes (here, for example, they areparallel). These include the spin-resolved transmission coefficients,spin polarized charge current, spin-current, spin resolved scatteringstates, spin transfer torque etc.

The calculator then changes the magnetic configuration of the electrodes(say to anti-parallel), and for this new magnetic configuration of thetwo electrodes (anti-parallel), the calculator starts again andconstructs an initial device Hamiltonian H, so that transport propertiesof the different magnetic configurations are obtained.

When devices with both parallel and anti-parallel 3D magneticconfigurations of the electrodes are simulated, the calculator thenobtains important device merits such as tunnel magnetoresistance ratio,spin injection coefficient, etc. It is very important to have bothparallel and anti-parallel 3D magnetic configurations done in aself-consistent manner as described, otherwise these important devicemerits cannot be obtained.

When necessary, other magnetic configurations (e.g., the two magneticmoments of the two electrodes are at an angle to each other) can bemodeled in similar fashion with an additional local rotation of magneticmoment.

The present invention has therefore a large potential application inseveral markets. On the one hand, it is most useful for researchers innanoelectronics, nanotechnology, device technology, informationtechnology, spintronics, quantum transport modeling, and materials. Onthe other hand, it is useful for R&D in magnetic storage and magneticdevice industry, magnetic sensors, and in spintronics, emerging as abranch of nanoelectronics industry.

The present method allows accurate modeling of magnetic atoms involvinglarger basis sets, small lattice constants, which means that large scalecomputation must be handled, which are efficiently parallelized to runin a parallel computing environment. The method provided is a verypowerful and versatile method based on the NEGF-DFT-LSDA/GGA formalismfor predicting spin polarized quantum transport.

As should now be apparent to people in the art, the present inventionprovides a modeling method based on atomistic and quantum mechanicalfirst principles, for quantitative modeling of spin-polarized chargetransport in magnetic devices. The method can be applied to calculatespin polarized charge transport driven by external electric fields, andprovides 1) a theoretical formulation in spin space so that thenon-equilibrium density matrix can be evaluated for different spinchannels; 2) a new way to perform very careful transverse momentumsampling for magnetic materials which serve as the device electrodes;and 3) an efficient parallelization of computing algorithms toaccurately model a large number of atoms.

As people in the art will appreciate, in contrast to Layer-KKR method,the present method is not limited for application in systems consistingof layers of materials and requires reduced computational time whileallowing modeling a bias and gate voltage.

As people in the art will appreciate, in contrast to LMTO method, thepresent method is not limited to bulk materials (or layered bulkmaterials) without external bias or gate voltage. The present methoddoes not rely on the atmospheric sphere approximation (ASA).

As people in the art will appreciate, the present invention, since it isbased on Green's functions, is extensible to many physical situationswhose theoretical analyses require Green's functions.

The present method allows performing ab-initio atomistic calculations ofspin polarized charge transport in realistic 3-dimensional devices underexternally applied electric fields.

Although the present invention has been described hereinabove by way ofspecific embodiments thereof, it can be modified, without departing fromthe nature and teachings of the subject invention as defined in theappended claims.

What is claimed is:
 1. A method for calculating spin polarized quantumtransport in a 3-dimensional nanoelectronic device under non-equilibriumconditions at finite bias voltage, comprising the steps of: a)self-consistently obtaining the Hamiltonian H of the device using DFTwithin the standard local spin density approximation for open devicestructures; b) constructing a non-equilibrium density matrix of thedevice using Keldysh non equilibrium Green's functions in spin space;and c) calculating spin-dependent transmission coefficients from theGreen's functions. wherein said step b) comprises a transverse momentumsampling of the Brillouin zone for contribution of each transverse Blochstate in directions perpendicular to the current flow for converging thedensity matrix.
 2. The method of claim 1, comprising incorporating oneof i) local spin density functional (LSDA) and ii) generalized gradientapproximation (GGA).
 3. The method of claim 1, wherein in said step a)DFT is applied using standard norm-conserving pseudopotentials andatomic orbitals are used as a basis set for electronic wavefunctions. 4.The method of claim 1, wherein said step c) comprises calculating thespin-dependent transmission coefficients from Green's functions as:$\begin{matrix}\begin{matrix}{{{T_{\sigma}\left( {E,V_{b}} \right)} = {\sum\limits_{k_{x},k_{y}}^{\;}{T_{\sigma}^{k_{x},k_{y}}\left( {E,V_{b}} \right)}}},{where}} \\{T_{T_{\sigma}}^{k_{x},k_{y}} \equiv {{Tr}\left\lbrack {{{Im}\left( \sum\limits_{L}^{r} \right)}G^{r}{{Im}\left( \sum\limits_{R}^{r} \right)}G^{a}} \right\rbrack}}\end{matrix} & (1)\end{matrix}$ is the transverse momentum resolved transmissioncoefficient.
 5. The method of claim 4, further comprising obtainingspin-polarized charge current as: $\begin{matrix}{{{I_{\sigma}\left( V_{b} \right)} = {\frac{e}{h}{\int_{\mu_{L}}^{\mu_{R}}{{{T_{\sigma}\left( {E,V_{b}} \right)}\left\lbrack {{f_{L}\left( {E - \mu_{L}} \right)} - {f_{R}\left( {E - \mu_{R}} \right)}} \right\rbrack}{E}}}}}{where}} & (2)\end{matrix}$ μ_(R,L) is the electrochemical potential of the left(right) leads and μ_(L)−μ_(R)=eV_(b); ƒ_(L,R)(E−μ_(R,L)) are the Fermifunctions.
 6. The method of claim 5, further comprising calculating thetotal charge current as: I≡I_(↑)+I_(↓).
 7. The method of claim 6,further comprising obtaining TMR from the total currents as:TMR=(I _(parallel) −I _(antiparrallel))/I _(antiparallel).
 8. A methodfor self-consistently computing non-equilibrium spin resolved densitymatrix for three-dimensional (3D) magnetic systems consisting of acentral scattering region comprising a non-magnetic material, a left anda right magnetic electrodes serving as device leads which connect thedevice to the outside world, comprising self-consistently calculatingthe Hamiltonian of each region within DFT in the local spin densityapproximation and generalized gradient approximation under a potentialincluding an applied bias potential V_(b) that drives the current flow;whereby the Hamiltonian corresponding to the left and rightferromagnetic electrodes are calculated as isolated bulk material andthe k_(∥)-dependent retarded self-energies of each lead are determined;and the Hamiltonian corresponding to the central region is calculatedself-consistently using the non-equilibrium electron density matrix,calculated over a 2D (in the x-y direction) Brillouin zone forcontributions of each transverse Bloch state; and spin currents arecalculated by integrating the contributions from each transverse Blochstate.
 9. The method of claim 8, wherein the magnetic materialelectrodes are one of magnetic atoms, alloys and compounds; and thenon-magnetic material is selected in the group consisting of a metal,molecules, semiconductors and insulating materials.
 10. The method ofclaim 9, wherein the magnetic atoms are one of Fe, Ni, Co.
 11. A methodfor calculating spin polarized quantum transport in a 3-dimensionalnanoelectronic device under non-equilibrium conditions at finite biasvoltage, comprising the steps of: self-consistently obtaining theHamiltonian H for an open device using DFT; obtaining the electronicstructure of the device by using NEGF to handle the nonequilibriumstatistics, and the device transport boundary conditions using a realspace numerical technique; and implementing local spin densityfunctional and generalized gradient approximation) for nonequilibriumNEGF-DFT calculation.
 12. The method of claim 10, wherein for DFT,standard norm-conserving pseudopotentials are used and numericalorbitals are used as a basis set.
 13. A computer system for calculationof spin-polarized charge current and spin current transport in magneticdevices; said system comprising: a calculator; and a database; wherein,said calculator computes the electronic structure of each 3D magneticelectrode of the device using DFT at equilibrium; and stores them insaid database; then, the calculator shifts the electrode potential by anamount determined by an external bias voltage; for a first magneticconfiguration of the electrodes: i) said calculator constructs a firstdevice Hamiltonian H from a first spin-resolved nonequilibrium densitymatrix and, using the first H, the calculator computes the NEGF of thescattering region of the device for the first magnetic configuration ofthe electrodes; ii) from the obtained NEGF, the calculator calculates asecond spin-resolved nonequilibrium density matrix; and using the secondspin-resolved nonequilibrium density matrix, the calculator constructs asecond device Hamiltonian; the calculator repeating steps i and ii)until the second device Hamiltonian H and the first device Hamiltoniandefer by only a pre-specified amount; then the calculator computestransport properties for the device in the first magnetic configurationof the electrodes; and then changes the first magnetic configuration ofthe electrodes to a second magnetic configuration of the electrodes andstarts again.
 14. A computer system for calculating spin polarizedquantum transport in a 3-dimensional nanoelectronic device undernon-equilibrium, comprising: a calculator; a controller; and a storageunit; wherein said calculator, from a periodic atomic structure in3-dimensional space of each electrode of the device, computes theelectronic structure of each electrode of the device; said storage unitstoring the electronic structure of each electrode of the device; andsaid controller then shifting the electrode potential by an amountdetermined by an external bias voltage; for a first magneticconfiguration of the electrodes, the calculator constructs a firstdevice Hamiltonian H from a first spin-resolved nonequilibrium densitymatrix; using the first device Hamiltonian H, the calculator computesthe NEGF of a scattering region of the device for the first magneticconfiguration of the electrodes; from the obtained NEGF, the calculatorobtains a second spin-resolved nonequilibrium density matrix; using thesecond spin-resolved nonequilibrium density matrix, the calculatorconstructs a second device Hamiltonian; until the second deviceHamiltonian and the first device Hamiltonian defer by a pre-specifiedamount; and then the calculator computes transport properties transportproperties for the device in the first magnetic configuration; then, thecontroller changes to a second magnetic configuration of the electrodes;and for the second magnetic configuration, the calculator starts again.